55.4 Akaike’s FPE, AIC, and BIC Criteria 
FPE. Akaike“ first proposed computing the final prediction error (FPE) in deter- 
mining the order n of AR(n) as 
N+q: 
FPE(q) = ——1-62 (5.252) 
N-q 
where N = number of observations to which the model is fitted, g = the order of the AR 
model 
62= {R(0) + a:R(1) +- + agk(g)| 
and adopting as n the value of q that minimize the FPE, as in Fig. 5.45. 
FPE (q) 
fe 
n 
Fig. 5.45. FPE vs. q. 
AIC. Later Akaike*!~* proposed a more refined method of minimum AIC 
(Akaike’s Information Criteria) based on information entropy theory. This is a very gen- 
eral concept available for general statistics problems, as an example of curve fitting, that 
is shown in Appendix A-2.*° AIC is defined as 
AIC(q) = (— 2) loge[Max. Likelihood] + 2¢ (5.253) 
where q is the number of parameters. Including G2, n + 1 parameters have to be 
estimated, so 
g=ntl. (5.254) 
The log. [maximized likelihood] is L in Eq. 5.233 or, omitting the constant, 
N-n 
1 
2 
log Oe ~ 202 Q. 
Here Q is defined as the sum of squares and from Eq. 5.246 
R 1 
a2 
Oc = N Q. 
N-n 
2 
Ignoring the second term, which is constant, and inserting it in Eq. 5.253 
Thus pes 
to N 
logde-—@. (5.255) 
196 
